Properties JEE adds

The rationalised NCERT chapter drops the properties of determinants, but JEE uses them to evaluate determinants without full expansion. They hold for any order; the product rule |AB| = |A||B| is in the text.

Transpose: |A′| = |A|, so every rule about rows also holds for columns.

Swapping two rows (or two columns) changes the sign of the determinant. So a determinant with two equal rows is 0, and so is one with two proportional rows.

A common factor of one row can be taken outside: multiplying a single row by k multiplies the determinant by k. Doing it to all n rows gives |kA| = kⁿ|A|.

Adding a multiple of one row to another, Rᵢ → Rᵢ + kRⱼ, leaves the determinant unchanged. This is the main tool for creating zeros before expanding.

If every entry of one row is a sum of two terms, the determinant splits into the sum of two determinants, one for each term.

A triangular matrix (all zeros above, or all zeros below, the leading diagonal) has determinant equal to the product of its diagonal entries.

Cramer's rule for |A| ≠ 0: x = ∆₁/∆, y = ∆₂/∆, z = ∆₃/∆, where ∆ = |A| and ∆ₖ is |A| with column k replaced by B. It gives the same answer as X = A⁻¹B.